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INDUCTANCE MODELING USING
NEW ELECTROMAGNETISM
by
Robert J Distinti
A thesis submitted in partial fulfillment of
the requirements for the degree of
Masters of ECE
Fairfield University
2007
Thesis advisors:
Dr. Jerry Sergent
________________________
Dr. Douglas A. Lyon
_________________________
Page 3
ABSTRACT
INDUCTANCE MODELING USING
NEW ELECTROMAGNETISM
By Robert J. Distinti
Improved electromagnetic field models, known as New Electromagnetism (NE)
[NE_INTRO], are applied to model the inductance of printed and integrated
circuit traces.
A review of Classical Electromagnetic theory (CE) shows many attempts to
quantitatively derive a model for circuit inductance. This thesis shows that these
derivations are either singular or incorrect applications of CE.
The Partial Inductance model (PI) is the current industry standard for modeling
inductance [Ruehli]. The PI model is both singular and contradicts CE in at least
two ways.
This paper derives a model for circuit inductance from NE which is free of
singularities; furthermore, it is simpler to apply than PI and is well suited for
adaptive numerical integration techniques which return accurate inductance
values with processing times that are at least an order of magnitude faster than PI.
Conductive planes (ground planes) significantly affect the inductance of a circuit
trace. NE provides a simple optical method to determine the inductance of a trace
suspended over a single conducting plane (single plane) and the situation where a
trace is sandwiched between two conducting planes (dual plane). CE applications
model the single plane as simply a return path [Holloway]. No consideration is
made for eddy currents induced in the plane or for the dual plane case. Microstrip
and stripline geometries do consider conducting plane effects; however, these
geometries are considered waveguides since the signal travels in the dielectric
medium between the strip and the ground plane [Pozar]. Waveguides are a topic
of follow-on work in NE.
Page 4
Table of contents
1. Motivation ....................................................................................................6
2. Historical Review.........................................................................................7
2.1.
Faraday’s law .................................................................................9
2.2.
The Neumann Equation ...............................................................11
2.3.
Partial Inductance (PI) .................................................................12
2.4.
Another Valiant Attempt using Classical Theory ........................16
2.5.
Alarm Bells ..................................................................................18
2.6.
Intrinsic inductance ......................................................................19
2.7.
The closed-path fallacy ................................................................20
2.8.
The Continuous Current distribution ...........................................20
2.9.
Wrap up of Historical Review .....................................................21
3. New Electromagnetism ..............................................................................22
3.1.
New Electromagnetism Definitions .............................................24
3.2.
Potential and Kinetic energy ........................................................25
3.3.
Potential and Kinetic voltage .......................................................26
4. The Circuit Definition of Inductance .........................................................28
4.1.
What is emf? ................................................................................28
5. Induction using New Electromagnetism ....................................................31
5.1.
Step 1: Determining the M-Field .................................................32
5.1.1.
The Hybrid Approach…………………………………….33
5.2.
The Seed.......................................................................................34
5.2.1.
Computing the seed value for copper…………………….36
5.2.2.
Developing the Continuous current model……………….40
5.2.3.
Mapping the M-Field……………………………………..42
5.3.
The Second Step: Integrate the M-Field to find the Kinetic Voltage 44
6. Comparison ................................................................................................46
6.1.
Comparing to Partial Inductance .................................................46
7. Ground Plane/Conducting Plane Effects ...................................................49
7.1.1.
Dual Conducting Planes………………………………….53
8. Validation ...................................................................................................57
8.1.
Characterizing the LCR Meter and Fixture .................................66
8.2.
Measurements ..............................................................................67
8.2.1.
LCR Correction Factor……………………………………69
8.3.
Experimental Data .......................................................................71
9. Processing Time Comparison ....................................................................77
10. Conclusion ...............................................................................................79
11. References ................................................................................................81
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1.Motivation
As the density and frequency of modern integrated circuits and printed circuits
increase, the need for accurate modeling of circuit properties becomes more
important. The inductance and capacitance of an integrated circuit (IC) trace is
governed by its geometry, material, parasitic effects from other traces and ground
planes. Any given trace could experience an unwanted resonance or “roll-off”
that, under certain conditions, could lead to an intermittent failure. Capacitive
and inductive coupling between traces is another source of intermittent failure.
Accurate modeling of circuit properties, prior to fabrication, reduces the number
of development “re-spins”. A “re-spin” is a re-design that requires new
prototypes to be produced. Reduction of the number of development “re-spins”
correlates to faster time to market, lower cost and improved reliability. Reducing
“re-spins” further reduces the environmental impact and cost overhead associated
with short production runs required to produce prototypes.
Eliminating the uncertainty and time associated with computer modeling of vast
networks of signals allows computer tuning of individual traces; thereby
eliminating the wasteful “one-size fits all,” “worst-case” termination methods.
Utilizing this technique, device designers can reduce the power consumption of
the end product; thereby, saving energy and the environment.
.
Page 6
2.Historical Review
According to CE, the inductance of a circuit trace (or wire) is the summation of
the self-inductance and intrinsic-inductance (A.K.A. internal inductance). There
are also “parasitic” effects from other traces and ground planes; these parasitic
effects are discussed elsewhere in this paper.
Before reviewing prior art, certain terms are defined for clarity. It is very
important to understand the subtle differences between the terms used by circuit
designers and those who study field theory. Circuit designers only have two
definitions for inductive components. The first is inductance (L) and the second
is mutual-inductance (M). Referring to Figure 2-1, inductance (L) relates the
back-emf in a loop to the current change through the same loop. Mutualinductance (M) relates the emf in a loop caused by the current change in another
loop.
L
Math
Symbol
M
P
P
di
emf   L
dt
Loop 2
Loop 1
emf 2   M
di1
dt
Figure 2-1: Induction terms used by circuit designers
Field theorists calculate L and M for the circuit designer based on three different
field interaction models. These three field interactions are intrinsic-inductance,
Page 7
self-inductance and mutual-inductance. The inductance of a circuit is the
combination of self-inductance and mutual-inductance as shown by the following
relationships.
L=self-inductance + intrinsic-inductance
M=mutual-inductance (same name is used for the field model)
SELF-INDUCTANCE
According to CE, the-inductance of a loop is caused by the time-rate-of-change of
the magnetic flux in the space circumscribed by the loop. The source of the
magnetic field is the current in the loop. The classical field model for selfinductance is Faraday’s law combined with the Biot-Savart model (all discussed
later).
INTRINSIC-INDUCTANCE
The other component of inductance is known as intrinsic-inductance. This
inductance is cause by the field energy inside the wire. It is developed using
conservation of energy techniques, ostensibly because Faraday’s law can not
explain it [Hayt].
Page 8
MUTUAL-INDUCTANCE
Mutual-inductance occurs when the changing magnetic field of one loop couples
to a second loop. This thesis does not discuss mutual-inductance in great detail, it
is proper to note that all of the classical models provide excellent results when
applied to mutual-inductance problems.
Now that necessary definitions have been covered, the discussion continues with
the review of prior-art.
Faraday’s law
2.1.
The most basic self-inductance problem is the single turn inductor as shown in
Figure 2-2. According to CE, the self-inductance of this loop is determined from
Faraday’s Law (2-1), and the Biot-Savart flux model (2-3) [see Hayt].
W
I
S=perimeter
A=Area
Figure 2-2
d
dt
dI
2-2) emf   L
dt
IdS  rˆ
2-3) dB  K M
2
r
2-1) emf   n
Page 9
In the above equations,  is the number of flux lines contained in (or linked by)
the loop; n is the number of turns in the loop; emf is the voltage induced in the
loop as a function of flux change; I is the current in the loop. L is the inductance
of the loop; K M is

; B is the flux density (Webbers per square meter); r is the
4
distance from a differential segment of wire to the point where the flux density is
being determined.
First, integrate (2-3) around the perimeters to compute the flux intensity to yield
2-4) B  K M

IdS  rˆ
r
S
2
.
Then substitute (2-2) into (2-1) and reduce. Allow n=1 since we are only
considering single turn loops. The result is
2-5) L 

I
.
To find  , we integrate (2-4) over the area circumscribed by the loop to arrive at
2-6)
 IdS  rˆ 
  dA .
2


r
AS

 ( I )   B  dA  K M   
A
This resolves to
2-7)
 (I )  
thus L  
The above results are obviously useless; a typical reaction to this result is to
assume that the modeling of an inductor as a filament is the reason why the
inductance goes to infinity. This is wrong for two reasons. First, the diameter of
the filament is not parameterized; as such, there is no opportunity to “divide by a
zero diameter”. Second, this filamentary technique gives very accurate answers
for mutual-inductance problems.
Page 10
This model works well for mutual-inductance because the distance (r) from the
primary to the area circumscribed by the secondary does not go to zero. In the
self-inductance case, the primary and the second are the same; therefore, r goes to
zero causing the division-by-zero event.
2.2.
The Neumann Equation
Another variation of Faraday’s Law is called the Neumann equation (2-8)
[Lorraine, Ruehli] which is derived from Faraday’s Law using Vector Magnetic
Potentials.
2-8)
dP  dS
r
S P
M  KM  
The Neumann equation is expressed in terms of the mutual-inductive coupling
between a primary and secondary loop as shown in Figure 2-3.
P=primary
r=distance between
differential lengths
dP and dS.
dP
I
r
dS
W=radius
A=Area
S=secondary
D=distance
between loops
Figure 2-3 Parallel Loops
The Neumann equation (like Faraday’s Law) yields an undefined answer when
applied to self-inductance modeling (P and S same loop). This is obvious since r
would go to zero if d goes to zero.
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2.3.
Partial Inductance (PI)
Partial Inductance (PI) is the most successful classical attempt to address the issue
of modeling circuit inductance [Ruehli]. As the name suggests, PI requires that a
conductive loop be partitioned into straight sections as shown in the following
diagram:
“Inductive Partitions”
A Rectangular Loop
4
5
3
2
6
1
7
Figure 2-4: Partitioning a Loop for Partial Inductance
Once the loop is “partitioned,” an “Induction matrix” is created which catalogs
the “Partial Inductance” coupling between each pair of segments.
 L p11
L
 p 21
 L p 31

LP   L p 41
 L p 51

 L p 61
L
 p 71
L p12
L p13
L p14
L p15
L p16
L p 22
L p 23
L p 24
L p 25
L p 26
L p 32
L p 33
L p 34
L p 35
L p 36
L p 42
L p 43
L p 44
L p 45
L p 46
L p 52
L p 53
L p 54
L p 55
L p 56
L p 62
L p 63
L p 64
L p 65
L p 66
L p 72
L p 73
L p 74
L p 75
L p 76
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L p17 
L p 27 
L p 37 

L p 47 
L p 57 

L p 67 
L p 77 
Each entry on the diagonal of the above matrix represents the self-inductance of a
given partition (the coupling of a partition to itself) where-as the off diagonal
elements represent the mutual coupling between separate partitions. Thankfully,
because of reciprocity, only half the mutual terms need to be computed
( LPij  LPji ). The entries in the matrix are computed using the following PartialInductance equation [Ruehli]:
2-9)
LPkm  K M
1
a K aM
cK cM

aK aM bK bM
dl K  dlM
daK daM
rKM
The Partial Inductance Equation
Figure 2-5 highlights the parameters of equation 2-9)
aM
dlM
Partition M
daM
LM=length
rKM
aK
Partition K
daK
LK=length
dlK
Figure 2-5:
Page 13
The total inductance of the circuit is found from the matrix using the following
equation:
n
2-10)
n
Lloop    S KM LPKM
K 1 M 1
Where SKM is a sign function (+/-1) which takes care of the sign of the coupling
between the sections.
The PI equation (2-9) is, by definition, only applicable to straight conductor
partitions; therefore, the modeling non-linear shapes, such as circles, require
synthesis using a number of short partitions. This correlates to a matrix of partial
products equivalent to the number of partitions squared.
Ruehli develops PI from Vector Magnetic Potentials (VMP). The use of VMP
requires integration around a closed loop; this is echoed in a well respected
classical text book “Engineering Electromagnetics” 4th edition, the author William
Hayt states the following on page 276:
“[The] Equation [for Vector magnetic potentials] may be written in
differential form, if we again agree not to attribute any physical
significance to the [result] until the entire closed path in which the
current flows is considered.”
Each term in the PI matrix is defined as the emf developed in a given partition due
to the magnetic field caused by the current change in another partition. By
definition, this would be the inductive coupling between two linear partitions.
These terms are not calculated with a closed-path integral; therefore, we can not
assume “any physical significance” to their meaning. This makes sense from a
classical standpoint since you need to know the magnetic flux enclosed by a
Page 14
conductor in order to determine a valid emf. Since a linear partition can not
enclose anything, the values obtained for the matrix are meaningless as proposed
by Hayt. Ruehli tries to explain this by saying that the loops close at infinity;
however, that would require integration to infinity and back in order to close the
loop integral. This is certainly not shown in the equations; therefore, PI is
inconsistent with CE.
Another small problem with the PI is the handling of the current at the corners of
rectangular loops. Because PI only models linear partitions, the current at the
corner of a rectangular loop is modeled as if it continues on in a straight path,
without making the turn around the corner. A more accurate model should
account for the current at the corners making a turn.
The PI model also omits the intrinsic-inductance from the result. According to
classical theory, circuit inductance is the combination of the self-inductance and
the intrinsic-inductance. One may argue that PI actually considers the intrinsicinductance due to the fact that it integrates across the volume of the inductor.
This would be a valid argument except for the fact that another volumetric
solution (shown in the next section) involving Vector Magnetic Potentials
includes intrinsic-inductance as part of the solution.
Finally, PI exhibits the same Division-by-Zero (Singularity) problem that plagues
all solutions trying to use CE. Ruehli, to his benefit, acknowledges the singularity
problem (division by zero) with the diagonal elements in the matrix. He also
presents a reasonable means to “Normalize” the diagonal elements. Models
where the denominator tends to zero are inherently unstable.
With the New Electromagnetism approach, there are no messy matrices, there are
no “singularity” problems, there are no extraneous “sign” functions, the corners of
Page 15
rectangular loops are properly modeled and finally, circular loops do not present
an explosion in complexity. Before we look at NE we must complete the review
of classical theory.
2.4.
Another Valiant Attempt using Classical
Theory
Another valiant application of classical theory attempts to use vector magnetic
potentials. This attempt is found in the text book Classical Electrodynamics by
John David Jackson [Jackson]. The derivation starts on page 181 and yields the
following closed form solutions (equations 2-11, 2-12) for the inductance of a
loop of dimension shown in Figure 2-6.
2-11)
  8a  
LSELF   0 a ln    2
  b  
2-12)
  8a  7 
L   0 a ln    
  b  4
b
a
Figure 2-6
Jackson first derives the expression for the self-induction of the loop (2-11) before
adding in the effects of intrinsic inductance to arrive at the complete answer
which is equation 2-12. With careful analysis of Jackson’s derivation of 2-11, it
is shown that this equation is not the self-inductance of a loop; rather, it is the
mutual inductance between two loops as shown in Figure 2-7 [ind_jackson].
Page 16
a
b
Figure 2-7
As a secondary confirmation, the Mutual-Inductance between the two loops in
Figure 2-7 is derived by Maxwell using the same techniques. Maxwell’s answer
is:
2-13)
  8a  
M   0 a ln    2 [Maxwell].
  b  
The slight difference between equation 2-13 and Jackson’s result (
2-12) is due to the fact that Jackson accounts for the intrinsic-inductance. To
demonstrate this, we compute the value for the intrinsic-inductance of a single
loop and add it to 2-13 to arrive at the same result.
The accepted model for intrinsic-inductance is
2-14)

8
(H/m)
[Hayt].
Since the length of the loop is 2a , we get a total intrinsic-inductance component
of
2-15)
a
4
Henries.
When this result is added to 2-13, we get the same answer as Jackson which is:
2-16)
  8a  7 
 .
  b  4
 0 a ln 
Page 17
In this section it is shown that Jackson’s model for a loop inductor is actually the
summation of the mutual-inductance between a pair of loops and the intrinsicinduction of a single loop. This is not a valid solution for the inductance of a
loop.
2.5.
Alarm Bells
Equation
2-12 is derived from Vector Magnetic Potentials and Green’s theorem to allegedly
compute the self-inductance of a loop. Vector Magnetic Potential is just an
abstraction of the Biot-Savart model, and Greens Theorem is simply a method
whereby the total flux linked by a closed loop can be found from a Vector
Magnetic Potential. Therefore, these techniques are mathematically identical to
using the Biot-Savart model and Faraday’s law to find self-induction.
If it is not possible to develop a sensible (non-undefined) answer to self-induction
from the basic models (Faraday’s Law, Biot-Savart), then it can not be possible to
develop a sensible answer from any abstraction of those basic models. A solution
that is only possible from abstracted techniques should have set off alarm bells
years ago; but it didn’t. Why not? Alarm bells would certainly ring if one were
to state that a certain electrostatic problem is only solvable using electric
potentials and not Coulomb’s Law. This is because electric potentials are derived
from Coulomb’s law and are thus limited to what Coulomb’s law can solve;
likewise, the Vector Magnetic Potential techniques for induction are derived from
the basic models (Biot-Savart, Faraday) and therefore should not be able to solve
anything more than the basic models.
Page 18
2.6.
Intrinsic inductance
As stated in the beginning of this chapter, intrinsic-inductance is the inductance of
the loop due to the field energy in the wire.
Ironically, there is no classical force model that allows us to calculate intrinsicinductance directly (this also should have set off alarm bells). Instead, a “backdoor” conservation of energy approach results in

Henries per meter for
8
cylindrical wire. This relation states that the intrinsic inductance is related to the
length of the loop and NOT the thickness of the wire.
It would be nice to have an experiment to either confirm or deny the validity of
the intrinsic-inductance component; however, this would require a stable, wellposed model of self-inductance; for which CE has yet to provide.
A simple validation experiment would measure a bunch of different length loops
of different wire thicknesses. Then a “meaningful” self-inductance model would
be used to subtract the self-inductance component from the measurements. If the
remainder correlates to wire length and not to thickness, then we could be
confident of the intrinsic-inductance effect; however, there is no meaningful selfinduction model.
There is the Partial-Inductance model which seems to yield reasonable results;
however, it does not consider intrinsic-inductance as part of the solution. This
clearly shows a lack of consistency among classical models.
Page 19
2.7.
The closed-path fallacy
Classical models do not work for self-inductance problems for a number of
reasons. One reason is that electromagnetic theory is plagued by a false notion
that a closed-path is required to calculate inductance. This is echoed by Weber’s
Caveat: “It is important to observe that the inductance of a piece of wire not
forming a closed loop has no meaning” [Weber].
A simple review of dipole antenna theory shows that a dipole antenna resonates.
For dipole resonance to occur; capacitance and inductance are both required;
therefore, the inductance of a piece of wire not forming a closed loop has to exist.
There are many other examples that suggest the existence of straight wire
inductance.
The existence of straight wire inductance implies the existence of a point-charge
(or discrete) model for inductance. This is the foundation of the New Induction
model which is part of New Electromagnetism.
2.8.
The Continuous Current distribution
A more significant dilemma is that classical models are hold-over’s from the
1800’s when current was considered “fluid like”, and continuous. Since the
discovery of the electron in the Early 1900’s, we have come to know current to be
corpuscular (or discrete) in nature; however, the old “continuous current” models
are still considered valid under all conditions.
Page 20
When considering a mutual-induction experiment, the distance between the two
loops (primary and secondary) is substantially larger than the distance between
mobile carriers (the discrete charges of interest); therefore, modeling the current
in the primary as a continuous current yields a good result.
On the other hand, when considering a self-inductance experiment, we can
imagine it as a mutual-induction experiment where the primary and secondary are
the same loop. In that case, the distance between the primary and secondary is
zero. This is the source of the singularity problem.
Ultimately what is needed is a point-charge (or discrete) form of induction
allowing us to determine the induction of a wire construct at the level of the
mobile carriers. New Induction provides such a model and is discussed later.
2.9.
Wrap up of Historical Review
Although Faraday’s law is used to explain the mechanism of self-induction in
terms of classical field theory; it provides an undefined answer when one attempts
to quantitatively derive a solution for self-inductance.
The Partial Inductance model provides limited success to the self-inductance
problem; however, this success comes at the price of violating classical theory.
Furthermore, Partial Inductance does not properly address ground plane effects;
as such, solutions to realistic PCB and IC trace inductances remain unanswered.
Page 21
3.New Electromagnetism
The New Electromagnetism models were developed under the premise that ALL
electromagnetic fields are generated by forcing charges to behave a certain way
and ALL electromagnetic fields are measured by observing the behavior of
charges; therefore, charges are the fundamental building blocks of
electromagnetic physics.
Since RLC circuits are modeled with second order differential equations, there
must be second order equations for electromagnetic fields. Secondly, since
charges are the most fundamental building block of electromagnetism (at the
electrical engineers level at least), then there must exist second order force
equations based on discrete charges. That is, there must be fundamental force
equations which relate charge position, velocity and acceleration.
New Electromagnetism is the expression of such force equations.
Table 3-1: The New Electromagnetism Equations
The Force Equations
K Q Q rˆ
F  E S2 T
r
F
K M QSQ T
r
F
2
Name
Coulomb’s Model
v T  rˆ v S  v S  rˆ v S  ( v S  v T )rˆ 
(newtons)
New Induction Model
 K M QS QT a S
r
Where: K M 
(newtons)
New Magnetism Model
(newtons)

1
, KE 
. Bold letters represent vector quantities.
4
4
These equations represent the Force affecting a “target” charge due to the
behavior and position of a “source” charge. The ‘S’ and ‘T’ subscripts in the
Page 22
above equations delineate properties associated with the source and target
charges. The vector r represents the vector from source to target in all equations.
QS
aS
r
vS
F
vT
QT
Figure 3-1: Two Body Diagram
Coulomb’s model relates force to charge position. Other than notational
differences, it is identical to CE.
The New Magnetism model relates force to charge velocities. The first and third
terms inside the brackets can be derived directly from F  QvxB and the BiotSavart model. The middle term is required to make the model match well known
experiments [Secrets]. This model differs from classical theory in that it teaches
us that magnetism is actually a spherical field phenomenon; not the transverse
field as depicted in the Biot-Savart model (the B-field).
The third model is New Induction which was found through computer search of
experimental data. New Induction shows that an accelerating source charge
creates a disturbance that expands spherically from the source. This expanding
disturbance imparts a force in the opposite direction on target charges that it
intersects.
Page 23
Both New Induction and New Magnetism are based on a spherical magnetic field
model. The magnetic field of classical theory, as described by the Biot-Savart
model, is a transverse (like a donut) shaped field because it does not provide for a
magnetic field longitudinally to a moving charge (see Figure 3-2); therefore, New
Electromagnetism is not a reformation of classical theory; it is a superset.
QT
r
F
vS aS
Biot-Savart
QS
New Magnetism
New Induction
Figure 3-2 New Magnetism is a Super Set of Classical field theory
New Electromagnetism is ideally suited for modeling induction simply because
the models reflect the electromagnetic interactions among discrete charges.
Lastly, the spherical field shapes are easier to model than the toroidal B-field of
CE; thereby, reducing processing time and code complexity.
3.1.
New Electromagnetism Definitions
The following definitions are required for the application of New Inductance (NI)
to modeling circuit inductance.
Page 24
The New Electromagnetism (NE) field models are developed by dividing the
Force models by the target charge. This yields two types of force-per-coulomb
fields as shown in the following table.
The Field Equations
E 
K E QS rˆ
r
M
Electric Field Model
2
K M QS
r
M
Name
2
v T  rˆ v S  v S  rˆ v S  ( v S  v T )rˆ 
(Newtons/Coulomb)
Magnetic Field Model
(Newtons/Coulomb)
Inductive Field Model
 K M QS a S
r
(Newtons/Coulomb)
In CE, a force-per-coulomb field is typically designated with an “E”; however,
that would imply an electric field which is conservative. In NE there are two
types of force-per-coulomb fields, the Electric field (E) and the Magnetic Field
(M). E (Electric or Electro-Potential) fields are developed from Coulomb’s
Model while M (Magnetic or Magneto-Kinetic) fields are developed from both
New Induction and New Magnetism.
E fields convey potential energy and are conservative.
M fields convey kinetic energy and are not conservative.
The M field is NOT the B-field of classical theory since the M-Field truly is a
“Force” field like the Coulomb field.
3.2.
Potential and Kinetic energy
New Electromagnetism provides for both potential and kinetic energy.
Page 25
Potential energy (PE) is developed only from the E field and is defined by the
following equation which is identical to classical theory.
b
3-1)
PE  Q  E  dL
a
This states that potential energy is acquired as a charge is moved against the
applied electric field. A negative result indicates potential energy lost to other
energy forms such as kinetic energy.
Kinetic energy is developed from both E and M field. Its is defined by the
following two expressions
b
3-2)
KE  Q  E  dL
a
b
3-3)
KE  Q  M  dL
a
These equations state that a charge acquires kinetic energy if it is allowed to move
in the direction of the applied field. A negative result indicates kinetic energy lost
to other forms. Other forms of energy include, but are not limited to, potential
energy or energy dissipated in the form of heat.
3.3.
Potential and Kinetic voltage
In modern text books, voltage is defined as the energy per coulomb; however,
potential energy is assumed in all cases (even in the case of emf). Maxwell
describes electricity in terms of kinetic and potential energies; however, the term
emf is used with both in a ubiquitous manner [Maxwell].
Page 26
New Electromagnetism defines two types of voltage, one for kinetic energy per
coulomb (see Equation 3-4), and another for potential energy per coulomb (see
Equation 3-5).
KE
= kinetic voltage (Joules per coulomb)
Q
PE
3-5) V P 
= potential voltage (Joules per coulomb)
Q
3-4)
VK 
The importance of kinetic and potential voltages is discussed in the next section.
Page 27
4.The Circuit Definition of Inductance
The ultimate goal of this work is to use field models to arrive at the inductance of
an inductor. To this end, experiments are performed to validate the predictions of
the field models; therefore, the discussion of inductance from the standpoint of
circuit theory is in order.
In circuit analysis, the inductance (L) of a circuit relates the “back emf” across the
inductor to the current change through it. This is shown in the following
expression.
4-1) emf   L
di
Circuit Relationship for Inductance
dt
If we then solve for L, then we can determine the Inductance from measurements
4-2) L   emf
di
Induction from Circuit measurements
dt
The above expression enables us to determine inductance by measuring the emf
across an inductor excited by a current change.
This begs the simple question: What is emf and how do we measure it?
4.1.
What is emf?
Modern texts book [Hayt] state that emf is voltage (or potential); however, this
definition leads to a paradox.
The paradox is demonstrated by returning to Faraday’s Law which states that a
loop of wire exposed to changing magnetic field produces an emf.
4-3) emf   n
d
Faraday’s Law
dt
Because emf is assumed to be potential, this equation is modified using Stokes’
theorem into one of Maxwell’s famous equations
4-4)   E  
dB
Maxell's version of Faraday's Law
dt
Page 28
This equation contradicts Kirchhoff’s voltage law which states that the voltage
around a close loop is zero. In point form it looks as follows
4-5)
  E  0 Kirchhoff's Voltage Law in point form
Again, texts books “dance” around this issue by stating the Kirchhoff’s Law is
only valid for “Static” field conditions [Hayt]. This is patently ridiculous since
Kirchhoff’s Law is used with great success in AC circuit analysis. AC circuits are
circuits where fields are certainly changing.
The proper resolution of this paradox requires us to redefine emf from magnetic
sources as kinetic energy per Coulomb as opposed to potential energy per
Coulomb. This notion is alluded to by Maxwell [Maxwell] in article 634 (and
others) where he explicitly states that the energy in a magnetic field is kinetic in
nature. By rewriting the above equation in New Electromagnetism terms we
have:
4-6) V K   L
di
Corrected Induction Circuit Equation
dt
4-7) V K   n
d
Corrected Faraday Law
dt
4-8)   M  
dB
Corrected Maxwell Equation
dt
With these changes, CE induction models (4-6, 4-7, 4-8) no longer violate
Kirchhoff’s Law (4-5); furthermore, Kirchhoff’s Law is valid under all field
conditions. Finally, since the M-Field of NE is not a conservative field, then in
NE, it is not a contradiction for a changing magnetic field to impart energy to a
closed loop of wire.
Again, we must be able to measure kinetic voltage in order to be able to perform
experimental validation of the theoretical results. All voltage measuring
instruments (volt meters, oscilloscopes, etc) only measure potential voltage.
The solution to this requires us to understand what happens to a loop of wire in a
changing magnetic field.
Page 29
Consider a hypothetical closed loop of perfectly conducting material. Suppose
the loop was exposed to a changing magnetic field characterized by a step
function in such a way that kinetic energy is imparted to the loop. After the field
change has settled, we are sure that kinetic energy has been imparted to the loop
since we can now detect a residual magnetic field emanating from the loop due to
a constant current flowing in the loop. A volt meter connected across any two
points of the loop will measure nothing since Coulomb’s model will ensure that
the distribution of charge around the loop is uniform in spite of the fact that it is in
motion.
If we were to break the loop, the kinetic energy of the current will be converted to
potential energy as the moving current “crashes” into the break. This will cause
charge depletion at one side of the break and charge concentration at the other
side of the break leading to a Coulomb field which can then be measured by a volt
meter. Because of conservation of energy, the potential voltage measured across
the break will equal the kinetic voltage added to the loop by the magnetic field
change.
This entire progression of events is identical to the operation of a Ballistic
pendulum where the kinetic energy of a bullet is measured by capturing it in a
bullet trap affixed to a pendulum. The kinetic energy of the bullet is converted to
potential energy as the pendulum swings against gravity.
Therefore, the kinetic voltage developed in each inductor experiment is measured
indirectly by measuring potential voltage developed across the inductor with the
assumption that the conversion between potential voltage and kinetic voltage
occurs fast enough to be considered instantaneous (for all practical purposes).
The exact procedures used for measuring for measuring the inductances of the
experiments are discussed in the experimental section.
Page 30
5.Induction using New Electromagnetism
This section details the step-by-step derivation of the inductance model for actual
inductors based on the point-charge form of New Induction.
To begin this derivation we start with an assumption.
ASSUMPTION: CONSTANT CURRENT DISTRIBUTION
The following derivation assumes that the current distribution in the inductor is
substantially uniform in the cross section and along the length. This assumption
is common to all prior-art reviewed. A more advanced application allowing for
non-uniform current distributions which is suitable for high-frequency inductors
is a follow-on work.
THE SOLUTION
Solving inductance for any arbitrary conductive form is divided into two steps:
The first step is to use the New Induction model to determine the M-Field caused
by a unit current change through the conductor. Only the M-Field present through
the volume of the conductor is required.
The second step is to perform a volume integration of the M-Field to determine
the total kinetic voltage. The total kinetic voltage per unit current change is by
definition the inductance.
Page 31
5.1.
Step 1: Determining the M-Field
The following discussion on the M-Field is applicable to conductors of all shapes
and sizes. For the sake of simplicity, we consider a simple rectangular length of
conductor as shown:
Current Change (dI/dt)
Current Density (J)
I
J
WH
Height (H)
dJ
1 dI

dt WH dt
Length (L)
Width (W)
Figure 5-1: Simple conductive form
The ideal solution to mapping the M-Field over the conductor is to pick a point in
the conductor; then sum the New Induction effect of all the mobile charges in the
conductor to that point as shown in the following diagram.
Current change
(I)
M(x,y,z)
Mobile Carriers
Figure 5-2: Using Discrete Model to compute the M-Field at a point inside conductor
This process is then repeated for each point where the M-Field is to be evaluated
until a sufficiently dense M-Field map is generated as shown in the next diagram.
Page 32
Current Change
Mobile
Carriers Not
Shown
M(x,y,z)
Figure 5-3: M-Field Map of conductive structure
The problem to this approach is that there are 18 coulombs of mobile carriers per
cubic millimeter of copper. This would take an inordinate amount of time to
compute an M-Field map.
New Electromagnetism does have continuous current models that are derived
from the discrete charge models; however, we would encounter the same divideby-zero problem that plagues CE. A practical solution is a hybrid approach.
5.1.1.
The Hybrid Approach
The hybrid approach utilizes the discrete charge models out to a certain distance
from the point where the M-Field is being evaluated (called the seed). Outside of
the seed, the continuous current models of New Electromagnetism are used.
Therefore the M-Field at any given point is the sum of the effects of the discrete
charges in the seed (the seed value) and the remainder of the conductor outside of
the seed evaluated using continuous techniques.
Page 33
The “Seed”
Current change
(I)
Continuous
model used
outside of “seed”
region.
Seed Size(S)
Discrete model
used inside
“seed” region.
M(x,y,z)
Mobile Carriers
Figure 5-4: Diagram showing seed usage
5.2.
The Seed
The “Seed” is the region immediately around the point where the M-Field is being
evaluated. Within the seed, the M-Field is evaluated using the discrete charge
model of New Induction. Using this form of New Induction, the Division-ByZero problem is avoided.
Definition: Seed Value. The seed value is the M-Field at the center of the
seed due to the mobile carriers in the seed.
An advantage of this approach is that the seed value only ever needs to be
computed once; this pre-computed seed value is then reused at each point where
the M-Field is to be determined. For all intents and purposes it can be treated as a
constant.
Care must be taken when using the pre-computed seed value since the placement
can affect the answer. The following table shows the seed value contribution
based on the position of the seed within the conductive material.
Page 34
Table 5-1: Seed Usage
At Interior
At Surface
Use Whole Seed Value
Use Half Seed Value
At Edge
At Corner
Use Quarter Seed Value
Use 1/8 Seed Value
Other configurations where
seed is partially exposed
Can’t use pre-computed
seed value.
In order to use the seed value, the sample spacing of the M-Field can be no
smaller that ½ the seed size (S=length of seed side). The reason for this is that
when moving from an edge sample to a surface sample, the closest surface sample
can be no closer than S/2 to the edge otherwise the seed will be partially exposed
in such a manner that the seed value can not be used.
Page 35
As a final note, it is OK to use M-Field spacing smaller than S/2 as long as any
partial exposure condition where the seed value can not be used is either avoided
(M-Field not considered there) or the seed contribution is computed specially for
that condition.
5.2.1.
Computing the seed value for copper
Computing the seed value for a given material requires knowledge of the
properties and structure of that material. Copper, along with many other
conductors, have a Face Center Cubic (FCC) structure. An FCC structure is
shown in the following diagram.
d
Figure 5-5: FCC structure
The FCC cell (shown by the red Cube) is aligned with its corners centered on 4
atoms. The arrangement causes atoms to be centered on each face of the cube
(hence the name). This yields a total of 4 atoms per cube (6 halves at each face
and 4 eighths at each corner). The size of the cell (d) is determined from the
following equations [FCC].
Page 36
d 3
AtomicMass  packing
Avagadro  density
Equation 5-1: Equation for length of an FCC cube
Where:
1) Packing is the number of atoms per structural cube (4 for FCC (copper))
2) Density is grams per cubic centimeter (copper=8.96 gm/cc)
3) Avogadro is Avogadro’s constant = 6.022141510e23 atoms /mole
4) AtomicMass is the atomic mass (copper =63.5463 gm/mole)
Result: d=3.612e-8 cm for copper.
Now that we have the dimension for an FCC cell, the next question is: how many
FCC cells should comprise the seed? The answer is a trade-off. Larger seed
sizes allow us to reduce the M-Field integration time since more of the answer is
pre-computed in the discrete part; therefore, there is less to do in the continuous
part. Conversely, by increasing the seed size we reduce the resolution of the MField (see the final note in the previous section). There may be other factors
governing selection of seed size such as the implementation of the algorithms that
one chooses.
The seed size chosen for this work uses a cube of 2001 FCC cells on a side (Ns).
The seed value of a cube of this size is just beyond the point where the addition of
another layer of cells (2003x2003x2003) does not affect the resultant seed value
by more than 0.1%. This eliminates the need to account for “half-atoms” along
the face of the seed. Odd Values are chosen because it was easier to write the
seed value Algorithm for an odd number of cells.
The computation of the seed value assumes that each atom in the structure
contains P mobile charges that can contribute to the seed inductance. P can be a
Page 37
fractional value; since, in many conductors the number of mobile carriers is not
the same as the number of atoms. Furthermore, not all mobile carriers are
excited when a given current is applied to the conductor; therefore, P is
constrained by the applied current. This will be covered in more detail in a
moment.
The M-Field value for the seed (seed value) is computed at a target atom at the
center of the seed. In New Electromagnetism, a charge can not couple directly to
itself; therefore, the mobile carrier around the center atom is the target which is
reacting to the M-Field; therefore, it can not be a contributor.
The M-Field value is computed at the center atom using a summation of the New
Induction Field model ( M 
 K M QS a S
) over the remaining atoms in the seed.
r
This is shown in the following equation:
total
5-2)
Qn a n
r
n 1
M center   K M 
n  center
Since P represents the quantity of active mobile carriers per atom, Q is replaced
by P. Also, since the density of current change is assumed to be uniform, then the
charge acceleration ( a n ) is the same for all active mobile carriers; therefore, the
equation is rewritten as follows:
total
5-3)
M center   K M Pa 
n 1
1
r
n  center
We next make the following definition
Page 38
total
5-4)
RawSeedValue  RSV  
n 1
1
r
n  center
The RSV can be computed using a number of simple algorithms. The RSV for a
Ns3 (2001x2001x2001) cell copper seed is
1.906919e7 (atoms/length).
The seed size (Length of side) is computed by multiplying Ns times the cell size
(d).
5-5)
S  dN S
Next we compute the value of P and the acceleration ( a ). We start by
considering that the seed is at the interior of a conductive form (such as a printed
circuit board trace).
Assuming a conductor with a rectangular cross section (HW) the current density
(J) entering the conductor is:
5-6) J 
I
HW
The amount of current that is passing through the seed (IS) is found by
multiplying the current density (J) by the cross sectional area of the seed (S*S).
5-7)
IS  I
S2
HW
(Current through seed)
Then we use the following identity which states that the current (I) along a length
of a conductor (L) is equivalent to a quantity of charge (Q) moving at velocity (v).
5-8) IL  Qv
(L and v in same direction).
Page 39
Using the side of the seed that is aligned with the conductor ( SL̂ ) for L and IS for
I yields the total charge motion in the seed related to the charge entering the entire
wire structure
5-9)
Qv  I
S3 ˆ
L
HW
Take the time derivative of both sides
5-10)
Qa 
dI S 3 ˆ
L
dt HW
Q is the total charge moving in the seed. We convert Q to P by dividing by the
number of atoms in the seed which is the total number of cells (N=Ns3=20013)
times the number of atoms per cell (4 for FCC), thus
5-11)
Pa 
1 dI S 3 ˆ
L
4 N dt HW
Substituting this into the equation for M yields
5-12)
M Seed   K M
1 dI S 3
ˆ Seed Value
[ RSV ]L
4 N dt HW
This result is set aside; the next objective is to compute the effect at the center of
the seed due to the rest of the conductor (outside the seed). This is done with the
continuous current form of New Induction.
5.2.2.
Developing the Continuous current model
Again, we start with the New Induction field equation
Page 40
5-13)
M
 K M QS a S
r
and the charge velocity identity
5-14) IL  Qv (L is a vector length).
Take the time derivative of identity (5-14) and substitute into (5-13) to yield
5-15)
M  K M
dI L S
.
dt r
Then take the derivative of (5-15) with respect to length to establish the quantity
of M-Field contributed by each differential length of the source (Ls). Also, drop
the s subscript so as not to misidentify this as the length of seed.
5-16) dM   K M
dI dL
dt r
We can consider a wire as a collection of filaments of cross section da; where the
current in each filament is J*da and J is computed the same as in the seed section,
thus
5-17) d M   K M
2
dI da dL
.
dt HW r
To compute M, we then perform volume integration and arrive at
5-18) M   K M
dI 1
dt HW

da dL
L a
.
r
Since this must only be performed exterior to seed, we add the following
restriction:
5-19) M ext   K M
dI 1
dt HW

L a
da dL
(a, L external to seed)
r
Page 41
To reiterate: Equation 5-19 is the component of the M-Field at the center of the
seed that results from current changes external to seed.
5.2.3.
Mapping the M-Field
The total M-Field is computed at any given point by summing the M-Field
contributed by the seed and the M-Field contributed by current changes external
to the seed.
5-20) Mtotal=Mseed+Mext
The following diagrams show the magnitude of the M-Field mapped over the
volume of length of conductor. Figure 5-6 shows the magnitude at the surface
(less the ends) and Figure 5-7 shows cuts through the volume at regular intervals.
The false color rendition represents maximum magnitude with the color white and
least magnitude with black (not necessarily zero).
Figure 5-6: M-Field Map of conductor surface
Page 42
Figure 5-7: M-Field map of cuts through conductor
Figure 5-8: M-Field map of rectangular loop with vanes showing field direction.
Page 43
5.3.
The Second Step: Integrate the M-Field to find
the Kinetic Voltage
Once the M-Field is known, the second step is to determine the total kinetic
voltage by integrating the M-Field. If the wire structure were filamentary, then
the kinetic voltage could be found with the following simple path integral:

5-21) VK  M  dL
Since a real inductor is not filamentary and the M-Field is neither conservative
nor uniform, we must perform volume integration over the M-Field to determine
the average kinetic voltage as shown in Figure 5-9 and equation
5-22.
VK   M  dL
L
dA
Current Change
A  HW
VK 
1
M  dL dA
A A L
Figure 5-9: Integrals Yielding Kinetic Voltage
5-22) V K 
1
HW
  M  dL dA
AL
Remember that L  
5-23) L  
VK
Thus:
dI / dt
1
1
M  dL dA Inductance from the M-Field
HW dI / dt A L
Equation 5-23 is the final step of the hybrid method for computing the inductance
of an inductor from NE. The first step is to compute the M-Field from equations
5-12, 5-19 and 5-20. The result of which is then inserted into Equation 5-23.
Page 44
Note: since the M-Field determined in the first step will be a discrete set of points
(a numerical process is assumed); then equation 5-23 will likewise need to be
adapted for numerical methods.
Page 45
6.Comparison
For the sake of comparison, allow the seed size to diminish to zero. This allows
equation 5-19 to comprise the entire solution for the M-Field.
Substituting (5-19) into (5-23) and reducing yields
6-1)
L  KM
1
HW 2
 dL m  dL n 
A L L  rKM dA
K M
Unlike the Hybrid Model described in the previous section, this model will result
in singularities; however, it puts us on the same footing as PI for the sake of
comparison.
6.1.
Comparing to Partial Inductance
If we were to substitute the same subscript notation as Partial Inductance into
equation 6-1, we arrive at:
6-2)
L  KM
1
HW 2
 
a K a M L K LM
 dL m  dL n 

daK daM
rKM


Then generalize it for any arbitrary cross sectional area
6-3)
L  KM
1
aK aM

 dL m  dL n
L L  rKM
K M
a K aM
The Continuous Current Model of New Induction
Then compare it to Partial Inductance
Page 46

da K da M

6-4)
LPkm  K M
1
a K aM
cK cM

aK aM bK bM
dl K  dlM
daK daM
rKM
The Partial Inductance Model
Partial Induction may look remarkably like the New Induction results; except that
the “absolute value” function around the dL’s makes this model more difficult to
apply. The absolute value of the dot product requires PI to include a separate sign
function that would not otherwise be necessary.
The New Induction result is the model for the WHOLE inductor, regardless of
shape or size; whereas the Partial Induction Model is only the solution for a linear
section of the loop since PI requires a non-linear inductor to be partitioned. The
following table breaks down the other comparisons.
Table 6-1: Comparison of New Induction to Partial Inductance
New Induction
Partial Induction
Singular
Types of
inductors
handled
Not if using the seed
Applies to any shape
inductor.
Treatment of
corners
Treatment of
circles or
curves
Proper handling of
corners
No change to basic
equation
Yes
Only applies to linear inductors.
Requires “partitioning” and matrix
operations to handle inductor shapes
more complicated than a straight
section
Corners are a source of error
Requires a synthesis using large
quantity if straight sections. Thus
requiring very large matrix.
Page 47
Extra
requirements
The seed methodology
(Chapter 5) prevents the
singularity
1)A sign function that relates the
sign of the partitions
2) Matrix reduction technique for
complex geometries
3) A normalization scheme to handle
the singularity issue
Scientific
The spherical field
Violates classical theory; closed
basis
models of New
loop requirements of VMP are
Electromagnetism
dropped, requires loops to close at
infinity but this is not done.
Intrinsic
Not needed, New
Although Ruehli does not mention it
Inductance
Induction is complete.
in his paper on PI, other researchers
account for it in their applications of
PI or other methods [Holloway,
Jackson, others].
Processing
Proportional to wire Proportional to the square of wire
time (see later volume
volume
section)
Single Ground Image method. Requires Considers the ground plane only as a
Plane
(see integration over the wire current return path [Holloway].
later section)
volume a second time.
Requires
integration
of
a
considerable volume of the ground
plane.
Dual Ground Image method. Requires No application found.
Plane
(see integration over the wire
later section)
volume many times
depending upon desired
accuracy.
Page 48
7.Ground Plane/Conducting Plane Effects
When multi-layer printed circuits are constructed, one or more of the layers are
typically used to distribute power or ground; as such, these layers are substantially
solid except for vias or perhaps a few odd traces that could not be fit in the
routing planes.
In some cases, conducting planes are inserted for shielding purposes; in other
cases, ground planes are required to ensure the proper impedances for stripline
and microstrip circuits.
A conductive plane severely affects the inductance of a PCB trace; therefore, any
practical application requires that these effects be taken into account.
For this section we use the term ground plane; however, the techniques described
in this section apply to any conductive plane regardless of whether it is connected
to a given potential or not.
Classical attempts to account for ground plane effects consider the ground plane
only as a return path [Holloway]. There is no provision to account for the eddy
currents induced in the ground plane or for the effects of a plane which is not part
of the circuit. The classical approach applies PI to the return current distribution
in the ground plane which leads to extensive computer processing time since a
volume substantially larger than the original trace must be integrated.
Furthermore, since the current distribution in the ground plane is not uniform,
many partitions are required which greatly expands the size of the inductance
matrix that needs to be generated and reduced.
Page 49
New Electromagnetism provides a simple technique to model the effects of a
conductive plane. This technique is similar to the method of images developed
for electrostatic charges [HAYT]. The difference is that the plane does not have
to be tied to a potential, nor does the plane have to be part of the circuit.
We begin by considering a quantity (Q) of charge accelerating over a conductive
plane (refer to Figure 7-1). We refer to this as the source charge for the sake of
discussion. According to New Induction, the accelerating source charge produces
an M-Field which forces the charges in the ground plane to accelerate in the
opposite direction. The accelerating ground charges in turn produce their own MField which in turn affects the source charge.
Accelerating charge (The Source Charge)
Currents induced in
ground plane from M
field of accelerating
charge
h=height above plane
Figure 7-1: Source Charge Accelerating Over Ground Plane
The question is: how do the induced currents in the ground plane affect the source
charge?
Through symmetry, we realize that a charge accelerating on the opposite side of
the plane induces the same ground plane currents (we are using the same
assumptions from the method of images of electrostatic theory [HAYT]). We
shall refer to this as the Image Charge.
Page 50
Currents induced in ground plane from M-Field
of “Image Charge”
-h
Charge on opposite side (Image
Charge) accelerating in same
direction as Source Charge.
Figure 7-2: Charge on opposite side (Image Charge) produces same ground plane currents
We observe that the ground plane currents are accelerating in the opposite
direction to either the source or image charge; therefore, we can remove the
ground plane as long as the direction (or polarity) of the image charge is reversed.
Source Charge
Source Charge
2h
2h
OR
Negative
Direction
Negative
Charge
Figure 7-3: Ground plane effects Simulated with Negative Image
Thus, the NE approach to modeling the effect of a ground plane is to use the
negative image of the current carrying construct. The technique is applicable to
high frequency modeling as long as propagation delays are accounted for [NIA1].
Page 51
Since copper is not a perfect conductor, we can’t use 100% of the image.
Experiments show that a 1OZ copper plane (1.35 mils thickness) has an
“inductive reflectivity” of 97% [NIA1].
The following rendering (Figure 7-4) shows the M-Field of three basic inductor
shapes without considering the ground plane effects. Notice that the circle and
rectangle are only half mapped, this is done to reduce processing time since, due
to symmetry, the other half will be the same.
Figure 7-4: M-Fields of three basic inductor shapes
The next rendering (Figure 7-5) shows the three basic shapes again except that
their negative ground plane images are enabled. Notice that the intensities of the
M-Field mappings are substantially reduced.
Page 52
Figure 7-5: The three basic shapes with ground plane images enabled
As a final note, this technique assumes that the ground plane extends sufficiently
beyond the shape and that there are no significant obstructions to current flow in
the ground plane (such as cuts in the copper).
7.1.1.
Dual Conducting Planes
If a circuit trace is sandwiched between two planes, it is not enough just to have
two images (one above and one below). Now the reflections of reflections must
be accounted for until the inductive reflectivity (97% for copper) diminishes the
effect [NIA1]. This is similar to the effect demonstrated in Figure 7-6 which
shows a candle situated between two mirrors.
Page 53
Figure 7-6: Reflections of Reflections
Figure 7-7: Rendering showing multiple images of circle and spiral
Page 54
For the inductor case, each time the image is reflected, it is negated and its
intensity reduced to 97%. This results in alternating image coefficients as shown
in Figure 7-8
(-0.97)3
(-0.97)2
-0.97
1
-0.97
(-0.97)2
(-0.97)3
Figure 7-8: Computing the coefficients for 3 pairs of reflected images
By definition, an image pair consists of the corresponding images on either side of
the original. Figure 7-9 shows how the induction algorithm converges as the
number of image pairs is increased. Because of the symmetry of the convergence,
it should be possible to determine the dual plane value from just a few images;
however, that task will be left for follow-on work. For now, dual plane values are
computed with 70 pairs of reflections.
Note: convergence will vary depending upon conducting plane material and the
distance from the circuit trace to the plane.
Page 55
Dual Plane Inductance for Large Rect 50Mil 1Oz
300
Inductance (nH)
250
200
150
100
50
0
0
10
20
30
40
Number of Image Pairs
Figure 7-9: Dual Plane Convergence
Page 56
50
60
8.Validation
To validate the New Induction solution, a range of various experiments involving
the following shapes were chosen:
Rectangle
Circle
A
A
H=Copper Thickness
H=Copper Thickness
B
W
W
D
D
C
C
Spiral: Center-lead
Spiral: Corner-lead
W
B
W
B
C
C
C
C
A
A
H=Copper Thickness
N=Number of turns
H=Copper Thickness
N=Number of turns
Figure 8-1: PCB inductor shapes and dimensions
The following table lists the experiments.
Page 57
Experiments
Exp
Thick
Shape
n
Oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
1oz
2oz
2oz
2oz
2oz
2oz
2oz
2oz
2oz
2oz
2oz
2oz
2oz
1oz
Large Rect
Large Rect
Small Rect
Small Rect
Spiral
Spiral
Large Rect
Large Rect
Small Rect
Small Rect
Spiral
Spiral
Large Circle
Large Circle
Small Circle
Small Circle
Large Circle
Large Circle
Small Circle
Small Circle
Large Rect
Large Rect
Small Rect
Small Rect
Spiral
Spiral
Large Rect
Large Rect
Small Rect
Small Rect
Spiral
Spiral
Large Rect
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
CP
Dimensions
A
B
C
mils
mils
mils
No
2600
3300 600
Yes
2600
3300 600
No
2300
2450 200
Yes
2300
2450 200
No
2150
2150 150
Yes
2150
2150 150
No
2600
3300 600
Yes
2600
3300 600
No
2300
2450 200
Yes
2300
2450 200
No
2150
2150 150
Yes
2150
2150 150
No
2400
300
Yes
2400
300
No
1800
100
Yes
1800
100
No
2400
300
Yes
2400
300
No
1800
100
Yes
1800
100
No
2600
3300 600
Yes
2600
3300 600
No
2300
2450 200
Yes
2300
2450 200
No
2150
2150 150
Yes
2150
2150 150
No
2600
3300 600
Yes
2600
3300 600
No
2300
2450 200
Yes
2300
2450 200
No
2150
2150 150
Yes
2150
2150 150
Dual
2600
3300 600
Table 8-1 Experiment dimensions
Page 58
D
mils
200
200
400
400
0
0
200
200
400
400
0
0
200
200
450
450
200
200
450
450
200
200
400
400
0
0
200
200
400
400
0
0
200
N
4
4
4
4
4
4
4
4
H
W
mils
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
1.35
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
2.7
1.35
mils
50
50
50
50
50
50
30
30
30
30
30
30
50
50
50
50
30
30
30
30
50
50
50
50
50
50
30
30
30
30
30
30
50
Dual Plane
Experiments
Thick
Shape
n
Oz
1oz
1oz
1oz
1oz
mils
mils
mils mils
Large Rect
Dual
2600
3300 600
200
Large Rect
Dual
2600
3300 600
200
Large Circle
Dual
2400
300
200
Large Circle
Dual
2400
300
200
Table 8-2 Dual Plane Experiment dimensions
33
34
35
36
CP
Dimensions
A
B
Exp
C
D
N
H
W
mils
1.35
1.35
1.35
1.35
mils
50
30
50
30
Note: In the above tables, blank entries represent dimensions that are not
applicable to the given geometry. For example, circles do not specify a B
dimension (see Figure 8-1)
The column ‘CP’ refers to conductive plane. Each loop is measured without
conducting planes, then again with a single conductive plane. Some loops are
measured with dual conducting planes as shown in Table 8-2.
Not all of the inductors are used in the dual plane measurements because the dual
plane techniques were not originally planned for this thesis. Spirals could not be
done because the test fixture does not have enough clearance to allow for the top
plane. The 2 oz experiments were omitted because the experimental results
(Table 8-4) show a small difference between 1oz and 2 oz traces. The large
circles and rectangles were deemed sufficient to demonstrate the dual plane
techniques. Follow-on research considers the dual plane techniques in more
detail.
The experiments were etched into 3 single sided 4x6 printed circuit board by a
professional outfit (Figure 8-2). One board contains the circle experiments etched
in 1oz (1.35 mils) copper. The other two boards contain identical rectangular and
Page 59
spiral experiments; the difference between these two boards being only the
thickness of the copper. One is etched in 1oz copper while the other is etched in
2oz (2.75 mils) cooper.
Figure 8-2: The Experimental Boards
To ensure accurate measurements and to reduce error cause by poor contacts, #6
brass screws were soldered to the pads (see Figure 8-3)
Page 60
Figure 8-3: Brass Screws used as terminals
An Agilent E4980A Precision LCR meter was obtained to gather the experimental
data (Figure 8-4). In order to ensure highly accurate and repeatable readings, a
special fixture was developed (Figure 8-5) to couple the experiments to the meter.
This fixture is compliant with the 4TP specification mentioned in the Agilent
Technologies Impedance Measurement Handbook [IMPEDANCE] section 3-2.
The LCR meter was purchased new and still shows the yellow calibration
traceability envelope attached.
Page 61
Figure 8-4: Agilent E4980A Precision LCR meter
¼ inch thick brass
shorting block
Figure 8-5: 4TP Test Fixture used for experiment
Page 62
In order to compensate for impedance of the test fixture, the LCR meter provides
for “Open” and “Short” compensation.
For “Open” compensation, the test fixture is left unconnected as shown in the
following figure. In the “Open” compensation mode, the LCR meters measures
the impedance of the test fixture at various frequencies and stores the information
internally. Essentially, the “Open” compensation nullifies the parallel capacitance
and leakage of the fixture from the measurements.
For “Short” compensation, a shorting block (shown in Figure 8-5) is attached to
the fixture. The shorting block consists of a ¼ inch thick brass flat stock with
terminal screws mounted similarly to the experiments. In the “Short”
compensation mode, the LCR meter measures the impedance of the fixture at
various frequencies and stores this information internally. Essentially, “Short”
compensation nullifies the series inductance and resistance of the test fixture from
the measurements.
Figure 8-6: Close-up of Test Fixture
Page 63
For experiments requiring a conducting plane (all of the even numbered
experiments) an un-etched PCB is inserted under the inductor being measured
(see Figure 8-7).
Figure 8-7: An un-etched 1oz PCB inserted underneath for measuring effects of single plane
In order to ensure that the inductor lays flat against the conducting plane, a
“Keeper” was constructed from wood and hot-melt glue as shown in Figure 8-8.
A thick text book was placed over the base of the keeper to ensure sufficient
pressure (not shown).
For dual plane measurements, a second single-sided un-etched PCB is placed on
top of the inductor being measured. It is oriented with the copper side up (see
Figure 8-9).
Page 64
Figure 8-8: The "wooden keeper" -- text books not shown
Figure 8-9: Dual Plane Measurement
Page 65
8.1.
Characterizing the LCR Meter and Fixture
This section highlights a procedure for evaluating the ability of the LCR meter to
compensate for the Test Fixture.
This procedure begins by allowing the LCR meter to warm-up for 1 hour. Then
both open and short compensations are performed. Following the compensation,
the Shorting Block is installed and a set of 32 measurements, at different
frequencies, are made. These measurements are plotted in Figure 8-10.
Since the LCR meter is not perfect there will be a residual inductance measured.
At the higher frequencies, the residual error is only a few Pico Henries. The LCR
meter was left running for 6 hours; after which, the same measurements were
taken. The second set shows that the residual error grew by a factor of 10.
At 2
MHz, the error was still less than 100 pH; therefore, as long as the measurements
are taken within 2 hours of compensation, we can expect instrument drift no
greater than 1 nH for frequencies 1 KHz and above.
Page 66
Inductance Readings with Shorting Block Installed
1000.000
After Short Comp
Inductance (nH)
100.000
6 Hours Later
10.000
1.000
0.100
0.010
0.001
0.01
0.10
1.00
10.00
100.00
1000.00
10000.00
Measurement Frequency (Khz)
Figure 8-10: Compensation and compensation drift
8.2.
Measurements
It would be preferable to take all the measurements at a single frequency;
however, due to frequency dependant errors (such as stray as capacitance) the
value obtained from the LCR meter may not be the actual inductance.
To explore a measurement strategy, measurements are taken at 32 frequency
points using the list mode of the LCR meter. The setup of the meter is shown in
Table 8-3.
Table 8-3: LCR meter settings used for experiments
Frequencies
Readings below 200hz are
not reliable, for the inductors
in this thesis.
200,300,500,750,1K,1.5K,2K,2.5K,5K
7.5K,10K,15K,20K,30K,50K,75K,100K,150K
200K,500K,600K,800K,1M,1.1M,1.25M
1.5M,1.75M,1.9M,2M
Page 67
Measurement Time
Long
(Aperture)
Averages
32
Mode
Ls-Rs
Compensation
Open and Short – All measurements taken no
later the 2 hours after compensation
All Other Settings
Default (power on)
Figure 8-11 plots the meter readings for experiments 11 and 12. The blue plot
shows the reading of the experiment without the ground plane and the magenta
plot is the reading with the ground plane. The yellow lines represent the
computer predictions for the given experiments.
It is interesting to note that below 1 KHz, the plots for both experiments merge.
This suggests that there is a certain M-field intensity required to knock electrons
loose in the ground plane. Below this threshold, the ground plane does not affect
the measurements.
After 100 KHz, both traces have an even downward slope which suggests that
other stray effects are beginning to dominate.
What is needed is a logical criterion to pick a measured value from the LCR meter
readings.
Page 68
(11,12) Spiral 30mil/1Oz
1200.00
1000.00
Inductance (nH)
800.00
L (nH)
L (nH) GP
600.00
Computer
Computer (GP)
400.00
200.00
0.00
100.00
1000.00
10000.00
100000.00
1000000.00
10000000.00
Freq (hz)
Figure 8-11: Measurement plot of experiments 11 and 12
Since the lower frequencies do not energize the conducting plane, then it is
reasonable to conclude that other stray effects are minimal as well. Therefore, the
measurements for experiments without conducting planes are to be declared
where the blue and magenta traces intersect (or where they come closest to
intersecting).
For the experiments with a ground plane, it would be preferable to choose the
measurement point at the same frequency as the sans ground plane experiment.
Since the ground plane is not energized at this point, we need to find a different
method.
8.2.1.
LCR Correction Factor
Since the intersection of the plots in Figure 8-11 is the desired measurement for
the sans ground plane experiments, then this knowledge is used to develop a
correction factor.
Page 69
We should like to take all measurements at 2 MHz since all ground plane effects
are energized and there is minimal instrument drift. However, since other stray
effects become more pronounced as frequency increases, then measurement error
is to be expected. Therefore, a correction factor is needed to offset these other
undesired effects (refer to Figure 8-12).
If the measurement at the intersection is what we define as the correct value for
the non-ground plane experiment, and the measurement at 2 MHz for the nonground plane experiment includes the correct value plus errors due to stray
effects, then the stray effects are found by subtracting the two measurements.
Stray_Effects@2Mhz = Measurement@2Mhz – Measurement@Intersection
The negative of the stray effects is the correction factor.
CorrectionFactor@2Mhz = Measurement@Intersection - Measurement@2Mhz
This correction factor must be determined for each inductor. All measurements of
this inductor made at 2MHz must be corrected by adding this factor. This
includes single and dual plane measurements.
Page 70
(11,12) Spiral 30mil/1Oz
No Ground Plane
1200.00
1100.00
Inductance (nH)
1000.00
Measurement at Intersection
900.00
Correction Factor
L (nH)
800.00
Measurement @ 2Mhz
L (nH) GP
700.00
Corrected Measurements @ 2MHz
600.00
500.00
400.00
100.00
With Ground Plane
1000.00
10000.00
100000.00
1000000.00
10000000.00
Freq (hz)
Figure 8-12: LCR measurement correction strategy
This correction factor is determined for each inductor shape being measured. The
correction factor is then applied to all measurements of that inductor (with and
without ground planes) to provide a measurement which is reasonably free of
stray effects.
8.3.
Experimental Data
Table 8-4 lists the computer prediction of a given experiment to the LCR meter
readings for experiments without ground plane. The experiment numbers
correspond to the experiment numbers found in Table 8-1.
Page 71
Table 8-4: Experiments without Ground Plane
Measured Computed
Exp #
(nH)
(nH)
%diff
1
274.0192
271.3814
1.0
3
223.9854
222.8745
0.5
5
1028.236
1029.098
-0.1
7
303.2772
299.5625
1.2
9
249.4881
247.4069
0.8
11
1101.615
1088.572
1.2
13
182.2881
176.6878
3.1
15
136.8191
133.4696
2.4
17
197.8299
195.2839
1.3
19
154.4495
149.1949
3.4
21
279.2265
269.9571
3.3
23
224.8992
221.6351
1.5
25
1034.499
1026.394
0.8
27
310.7126
297.2776
4.3
29
253.9133
245.4195
3.3
31
1110.758
1084.12
2.4
Figure 8-13 plots the computed value verses the measured values for each
experiment in Table 8-4. The red line drawn through the chart represents the
ideal line. If the measured value and computed value for a given experiment
matched, then the plotted point will fall on the red line. Error in either measured
or computed values will drive the experimental points farther from the red line.
Figure 8-13 shows excellent agreement between computer and measured values.
Page 72
Experiments w/o Ground Plane
1200
y = 0.9874x
R2 = 0.9997
Computed Values (nH)
1000
800
Coefficient of
Determination
600
Spirals
Average Abs Error =1.9%
Std Deviation of Error =1.3%
400
200
Single Turn Loops
0
0
200
400
600
800
1000
1200
Measured Values (nH)
Figure 8-13
Table 8-5 lists the computer predictions and LCR meter readings for experiments
with a ground plane. Figure 8-14 plots the computed value verses the measured
values for each experiment in Table 8-5.
Table 8-5: Experiments with Ground Plane
Measured Computed
Exp #
(nH)
(nH)
%diff
2
132.8018 141.1919
-6.3
4
117.0898 121.6218
-3.9
6
395.1197 371.9067
5.9
8
161.6237 169.2292
-4.7
10
142.992
146.0494
-2.1
12
475.773
431.0998
9.4
Page 73
14
16
18
20
22
24
26
28
30
32
103.5605
77.10436
108.2925
94.19904
139.2673
119.2943
398.2248
165.915
144.1099
470.8806
94.09387
76.95042
112.4716
92.52247
140.1886
120.7633
370.735
167.3749
144.4133
428.1903
9.1
0.2
-3.9
1.8
-0.7
-1.2
6.9
-0.9
-0.2
9.1
Experiments W/Ground Plane
500
450
y = 0.9393x
R2 = 0.9922
Computed Values (nH)
400
350
300
250
200
150
Average Abs Error =4.1%
Std Deviation of Error =3.3%
100
50
0
0
50
100
150
200
250
300
350
400
450
500
Measured values (nH)
Figure 8-14
Figure 8-14 shows that the data taken with ground plane does not correlate as
well as the non-ground plane experiments. This due to the following factors:
1) The addition of the ground plan increases parasitic capacitance which
affects the measured result. This is more severe for the spirals since the
Page 74
traces have significantly more surface area. This makes the inductance
more difficult to measure accurately.
2) The ground plane reduces the overall inductance by more than 50%;
therefore, the measurements are closer to the instruments noise margin.
One must remember that the computer predictions are based on a uniform current
distribution (a simplification). Even with this simplification, accuracy is obtained
to within 5% for non-ground-plane experiments and 10% or better for
experiments with ground plane. These are certainly useful tolerance values since
inductor manufacturers, such as Toko, offer standard tolerances of 5% and 10%
[TOKO].
Table 8-6 contains the dual plane results and Figure 8-9 shows the data plotted.
Table 8-6: Dual Plane Experiments
Exp #
Measured Computed
33 112.2639
114.6488
34
140.6
141.8519
35
77.36
76.33011
36
92.67
94.26503
%diff
-2.1
-0.9
1.3
-1.7
Page 75
Experiments:Dual Ground Plane
160
Computed Values (nH)
140
y = 1.0109x
2
R = 0.9978
120
100
80
60
40
20
0
0
20
40
60
80
100
Measured Values (nH)
Figure 8-15: Dual Ground Plane Results
Page 76
120
140
160
9.Processing Time Comparison
This section compares the processing time between the NE algorithm and the PI
algorithm.
Only straight conductors of varying length are considered. This allows us to use
PI without the need to break the conductor up into partitions.
Figure 9-1: Straight Conductor
The conductors have a width of 30 mils and height of 2 mils. The length starts at
10 mils and continues to double until either process breaks a ½ hour time limit.
The following chart compares the processing times.
Page 77
Processing Time for Straight Wire runs
4000
3500
Partial Induction
3000
New Induction
Seconds
2500
2000
1500
1000
500
0
0
200
400
600
800
1000
1200
Length of Run (mils)
Figure 9-2: Processing Time Comparison
Since the height and width of the conductors are held constant, then the length is
proportional to the volume of the conductor. This is important because both
algorithms are volume integrals which mean that processing time is a function of
volume.
The processing time for the NI algorithm is nearly linear with respect to wire
volume. The processing time for PI increases by the square of wire volume.
Therefore, the NI algorithm is nearly an order of magnitude faster than the PI
algorithm.
There is further room for improvement in the NI algorithms. For example,
expanding the seed size would reduce the integration time since more of the
integration is pre-computed.
Page 78
1400
10.Conclusion
This thesis demonstrates the modeling of inductance using New
Electromagnetism models and techniques. This is a preliminary work which
simplifies the modeling by assuming that the current distribution in a conductor is
uniform in cross section and along the length. This simplification is common to
prior art reviewed; and gives reasonably good results for low frequency
predictions as demonstrated by the excellent agreement between the theoretical
and experimental results.
Also demonstrated is a simplified approach to modeling ground plane effects for
both single plane and dual plane cases. This technique employs a negative image
of the circuit trace current. The image appears once for a single plane
implementation and multiple times for the dual plane implementation. This is a
more complete and simpler approach than the PI method [Holloway] which only
considers the single plane case in a return path configuration. The NE approach
produces results which are consistent with experiment.
This thesis further shows that all valid derivations of inductance based on CE are
singular. Those derivations that are not singular are shown to be erroneous. PI is
the industry standard model; ironically, it is both singular and in violation of CE.
The NE approach provides for numerical integration algorithms that are at least an
order of magnitude faster than PI. Furthermore, since the NE approach is not
singular, it is therefore inherently stable.
The next phase in this research involves the modeling of non-uniform charge
motion in a conductor. This will require the use of all New Electromagnetism
models to provide predictions of circuit properties which include capacitance,
Page 79
inductance, skin-effect and others. One of the goals being the ability to model a
circuit geometry with enough precision and detail to predict the LCR meter
readings at any frequency and mode. The ultimate goal being broad spectrum
modeling of circuit dynamics to include transmission line modeling and antenna
performance.
Page 80
11.References
[FCC] The following internet sources were used for the section on copper
structure http://www.ndted.org/EducationResources/CommunityCollege/Materials/Structure/metallic_stru
ctures.htm and
http://www.science.uwaterloo.ca/~cchieh/cact/applychem/metals.html
[Hayt] Hayt, William H., 1981. Engineering Electromagnetics, Fourth Edition.
McGraw-Hill Inc., New York.
[Holloway] Holloway, Christopher L. et al., 1998. Net and Partial Inductance of
a Microstrip Ground Plane, IEEE transactions on electromagnetic compatibility,
Vol. 40, No. 1.
[IMPEDANCE] (no Author listed), Agilent Impendence measurement hand book.
July 2006 http://cp.literature.agilent.com/litweb/pdf/5950-3000.pdf
[ind_jackson] Distinti, Robert J., 2004. A classical Attempt at Induction,
www.distinti.com/docs/ind_jackson.pdf.
[Jackson] Jackson, John David, 1999. Classical Electrodynamics, Third Edition.
John Wiley and Sons Inc., Hoboken, NJ. On page 181 is found a solution to a
current ring using Vector Magnetic Potentials.
[LCR1] (no author), November 2006. Agilent E4980A LCR meter users manual.
Third Edition. http://cp.literature.agilent.com/litweb/pdf/E4980-90020.pdf
Page 81
[Lorraine] Lorrain, Paul, 1988. Electromagnetic Fields and waves, Fourth
Edition. W.H. Freeman and Company, New York.
[Maxwell] Maxwell, James Clerk, 1891. A Treatise on Electricity And
Magnetism, Unabridged Third Edition, Dover. New York. This is a two volume
set first printed by Dover in 1954.
[NE_intro] Distinti, Robert J., 2005. New Electromagnetism Introduction.
www.distinti.com/docs/ne_intro.pdf
[NI] Distinti, Robert J., 1997. New Induction, www.distinti.com/docs/ni.pdf
[NI_neumann] Distinti, Robert J., 2003. New Induction and the Neumann
Equation, www.distinti.com/docs/ni_neumann.pdf
[NM] Distinti, Robert J., 2001. New Magnetism, Area-46 Publishing, Fairfield,
CT. www.distinti.com/docs/bk_000.pdf.
[Pozar] Pozar, David M., 2005. Microwave Engineering, Third Edition. John
Wiley & Sons, Inc. Hoboken, New Jersey.
[Ruehli] A. E. Ruehli, Inductance Calculations in a Complex Integrated Circuit
Environment, IBM Thomas J. Watson Research Center, Yorktown Heights, New
York.
[Secrets] Distinti, Robert J., 2004. The Secrets of QvxB,
www.distinti.com/docs/the_secrets_of_qvxb.pdf
[TOKO] (no author), Toko wire wound chip inductor catalog.
http://www.toko.co.jp/products/pdf/inductors/llm3225.pdf
Page 82
[Weber] Weber, E., 1965. Electromagnetic Theory, Dover, New York.
Appendix A: Material Constants
Material
Charge Mobility
Conductivity
Charge Density
Copper
m2
e
Vs
0.0032
1
m
5.80x107

Q
e m3
1.8125x1010
Aluminum
0.0012
3.82x107
3.1833x1010
Silver
0.0056
6.17x107
1.102x1010


Appendix B: Glossary
1oz
Thickness of copper equal to 1.35 mils
2oz
Thickness of copper equal to 2.75 mils
4TP
Four Terminal Pair lead configuration. It is the best method
according to the Agilent Impedance Measuring Handbook.
A
Cross sectional Area of conductor.
B
Magnetic field intensity in Webbers per square meter.
(classical theory only)
CE
Classical Electromagnetic field theory.
emf
Electro motive force.
Page 83
FCC
Face Center Cubic. This is the manner in which copper atoms
arrange themselves in metallic copper. In fact most conductors
have this arrangement.
H
The Height of the copper trace.
I
Current in Amperes.
IC
Integrated Circuit
J
Current Density in Amperes per unit area.
KE
Kinetic Energy
KE
Electric field Constant
KE 
KM
4
Magnetic field Constant
KM 
L
1

4
L = inductance
L = vector unit of length
LCR
Inductance Capacitance Resistance
M
M = mutual inductance
M = vector M-Field
mil
A unit of length equaling 0.001 inches
N
Total number of FCC cells in a seed
NE
New Electromagnetism
NI
New Induction
NM
New Magnetism
Ns
Number of FCC cells on the side of the seed
Page 84
P
The number of moving charges per copper atom (typically a
fractional number)
PCB
Printed Circuit Board
PE
Potential Energy
PI
Partial Inductance Model
Q
A quantity of charge in coulombs
RLC
Resistance Inductance Capacitance
RSV
Raw Seed Value
S
The length of one side of the seed
Seed
A small section of a conductor where the inductive effects are
computed at the atomic level.
Source Charge
The charge that is creating the field that is being studied
Target Charge
The charge that is reacting to the field that is being studied
VMP
Vector Magnetic Potentials
W
Width of a PCB or IC trace
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